Advertisements
Advertisements
प्रश्न
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
Advertisements
उत्तर
`vec"OA" = hat"i", vec"OB" = hat"j", vec"OC" = hat"k"`
D is the midpoit of BC
∴ `vec"OD" = (vec"OB" + "OC")/2`
= `(hat"j" + hat"k")/2`
E is the midpoint of AC
`vec"OE" = (hat"i" + hat"k")/2`
F is the midpoint of AB
∴ `vec"OF" = (hat"i" + hat"j")/2`
Now the medians are `vec"AD", vec"BE"` and `vec"CF"`
(i) `vec"AD" =vec"OD" - vec"OA" = (hat"j" +hat"k")/2 - hat"i"`
= `-hat"i" + hat"j"/2 + hat"k"/2`
`|vec"AD"| = sqrt(1 + 1/4 + 1/4)`
= `sqrt(1 + 1/2)`
= `sqrt(3)/sqrt(2)`
and d.c's of `vec"AD" = ((-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)))`
= `(- sqrt(2)/sqrt(3), sqrt(2)/(2sqrt(3)), sqrt(2)/(2sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/(sqrt(2)sqrt(3)), 1/(sqrt(2)/sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/sqrt(6), 1/sqrt(6))`
`["Now" sqrt(2)/sqrt(3) = sqrt(2)/sqrt(3) xx sqrt(2)/sqrt(2) = 2/sqrt(6)] = (- 2/sqrt(6), 1/sqrt(6), 1/sqrt(6))`
(ii) `vec"BE" = vec"OE" - vec"OB"`
= `(hat"i" + hat"k")/2 -hat"j"`
= `hat"i"/2 - hat"j" + hat"k"/2`
`|vec"BE"| = sqrt(1/4 + 1 + 1/4)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"BE" = ((1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2))) = (1/sqrt(6), (-2)/sqrt(6), 1/sqrt(6))`
(iii) `vec"CE" = vec"OF" - vec"OC"`
= `(hat"i" + hat"j")/2 - hat"k"`
= `hat"i"/2 + hat"j"/2 - hat"k"`
`|vec"CF"| = sqrt(1/4 + 1/4 + 1)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"CF" = ((1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)))`
= `(1/sqrt(6), 1/sqrt(6), (-2)/sqrt(6))`
APPEARS IN
संबंधित प्रश्न
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Write the distance of the point P (x, y, z) from XOY plane.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
Find the distance of the point (2, 3, 4) from the x-axis.
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
Find the direction cosines of a vector whose direction ratios are
0, 0, 7
If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `vec"a", vec"b", vec"c"`
If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
Find the direction cosine of a line which makes equal angle with coordinate axes.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
